I’ve only just discovered MathsJam, which may be meeting at a pub near you on the last but one Tuesday of each month, with Tuesday 13th., December a pre-Christmas exception. Here’s the MathsJam twitter site and a list of cities and other places where MathsJam is probably currently active; my personal centre is Nottingham, meeting at the Crafty Crow, opposite the Castle gateway.

Each month, Katie Steckles co-ordinates and issues a MathsJam SHOUT – a page of problems and puzzles to break any ice at your meeting. Here is a recent example and here is another. (Those two links are to dropbox – they did eventually appear for me without attempts to sign in with mis-remembered details and I did manage to print them off as A4 sheets.) The October sheet occupied me happily for an hour or more on a broken-down train to Nottingham for a meeting I consequently never made.

The annual MathsJam Conference at Yarnfield Park, Staffordshire, was a great treat. More than fifty talks were presented over the two days – each limited to just FIVE MINUTES after which escalating audio penalties are applied, and with just ONE MINUTE set-up time. After a batch of six or seven talks there is a coffee break during which the recent lecturers remain available to talk to. Delegates seat themselves at round tables, many strewn with mathematical games and puzzles, which makes for a very sociable time. Under the windows were arrays of free books, craft exhibits, a T-shirt competition, mathematical cakes competition, activities and puzzles competition,a competition for the best competion breaking the competition rules and arching over all a competition for the best competition. Next year we are threatened with a competition for the best best of competitions competition.

I think I’ve got that right. There was lots of laughter, generated by a very un-nerd-like array of stand-up mathematicians of all ages. After an excellent serve-yourself and seat yourself dinner there was an evening of activities, a quiz and mathematical musical jam session..

give you a flavour of that. Summaries and slides of the talks at the 2015 conference are already online and those for 2016 will be posted soon. Here are a few pictures from 2016, including the answers to an anagram (or was it an acronym) competition.

Thanks to the organisers, presenters, moderators, lift-givers, microphone-fitters and all who made MathsJam 2016 go so far as I know without a serious flaw. Finally, of course, here are my tessellations for the two days of the conference, November 12th. and 13th., Chinese Labyrinth (11,12) and Serpentine Labyrinth (13,11) each constructed via non-standard missing-links graphs in a quest for more intricate and less swastikoid forms. In the second example the illustrated missing links graph is itself swastikoid. Finding such graphs that work is helped by some experience and requires some “tweaking”.

Tomorrow, October 15th. 2016 is the 21st. annual Apple Day on Scarthin Promenade. The first was on October 21st, 1995, but we missed a year when it coincided with one of my many decadal birthdays. Until about five years ago it was run by Scarthin Books, but nowadays the energetic Celebrating Cromford team have taken over most of the work and the range of entertainments and instruction that accompanies the serious business of fruit pressing has grown. It feels almost like a harvest festival and for a long time very year has seemed to be the best ever.

It so happens that one supertile of Chinese Labyrinth (15,10), generated employing a fairly ingenious non-standard missing links graph fits, on my 44 x 44 squares, 2.4m x 2.4m tileyard, so I have been obliged to construct this celebratory tessellation. Here it is, seen whilst under construction.

and here is a fully-frontal:

Thank you to Eve Booker for the photographs taken in the gloaming. There’s more to be done tomorrow if time – I was called in for dinner.

More was done to modify the design – but not by me.

Here are some pictures of previous Apple Days.

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It’s hard to stop adding memorable and heart-warming images, and yes, the sun does often shine for us….touch wood, the forecast for tomorrow is a bit iffy.

The 2016 Wirksworth Festival, Derbyshire, UK, begins on Saturday 10th. September with the pioneering and now widely-famed Art and Architecture Trail which takes you from surprise to surprise, in and out of gardens, up and down the ancient hillside town temporarily or permanently colonised by artists from all over the town and from far and near.

This year the trail includes a half-foot-sized tiled realisation of a version of Chinese Lattice Labyrinth (9,8), chosen to celebrate the patronal (or matronal?) festival of St Mary the Virgin, patron of Wirksworth’s mediaeval St Mary’s Church, which fell as recently as yesterday, as I write, September 8th. Four supertiles of this tessellation have a wingspan of 44 squares, which, if sized two-by-two-inches, can just be fitted onto two 8 foot by 4 foot (approx 2.4 metres by 1.2 metres) plywood boards. Here’s the design, with added doodles, submitted for the approval of the Trail curators.

and in a rare moment of self-revelation here I am screwing tile-separators to one of the boards at 54 mm intervals (+ or – 1mm, I hope, or the 47mm x 47mm tiles won’t fit).

and here is the highly-skilled, self-moulded philosopher Jacob the Joiner (Jacob Butler), cutting up 1500 or so tiles on his rock-solid and millimetre-precise bench/table saw. It took only about three hours. The tiles are medium density fibreboard which when sawn splinters much less than real wood.

Other curiosities of the installation can be demonstrated if you dip into the Trail (completing it might take the two days) after purchasing a guide and badge at one of several accessible locations in the centre of Wirksworth. On a damp evening, Friday 9th. September, the completed piece was assembled by myself and son Michael, laid out flat on the lawn behind the Memorial Hall on St.John’s Street, necessarily sheltered by a substantial but obligingly pop-up gazebo.

The whole concept has worked but I’m a bit disappointed by its appearance; tomorrow we’ll perhaps add some parts of some further supertiles in pale blue (against the red) and charcoal grey (against the yellow) to achieve more contrast with the board.

PS now we’ve made this beast, its parts can be reassembled anywhere and to designs tailored to other dates or significant numbers. All rush at once.

The Lattice Labyrinths workbook is available from the publisher or you-know-who , or from a good independent bookshop , via Google or for two days only on the lawn behind the Memorial Hall in Wirksworth, where two striking giftwrap designs, “Logistical Nightmare” and “Diamonds are Forever” (oops – a copyright infringement, I take back that title), are also to be had at £1.50 a sheet, 10% to the Festival.

Bill Bryson remarks somewhere that a Martian visitor desiring to be taken to our leader would chose Vienna as the likeliest place to find him or her, judging by the magisterial grandeur of its imperial architecture and the overbearing opulence of the baroque sculptures that strut their muscles and bosoms along the Ringstrasse. Surely other capitals, in particular Saint Petersburgand Washinghton D.C.,would be major contenders, scale-model London being a 100:1 outsider. Whatever the Graeco-Roman grandeur of other capitals, however, I think that the archetype they all seek to emulate is PARIS.

So, on or around Bastille Day, it is time for a typically Gallic histrionic, over-the-top, ruthless and megalomaniac makeover of one of the city’s great public open spaces. The upset will be as nothing to that occasioned when les grandes Places et les Boulevards were originally butchered through the homes, workshops and lives of a teeming city, municipal destruction in scale and senselessness commensurate with British “slum clearance” of the 1950’s and 1960’s , except that little very impressive came out of the latter.

Le Quatorze Juillet is the number pair (14,7), is of form one even, one odd, so yielding a Lattice Labyrinth tessellation of the Chinese family, specifically of a subfamily where the two numbers share a common factor,in this case 7.

Here is a little region of Chinese Lattice Labyrinth (14,7).

The outline of this figure does display that cockiness (suffisance?) which forms part of the French national self-image and can arouse animosity among les étrangers.

Below is the present unfinished state of the re-paving of the Place de la Concorde; batches of slabs in the national colours have to be specially manufactured.

After many blog posts inspired by dates, including the Queen’s Birthday, Christmas Day, the National Days of Argentina, Brazil and Taiwan and the date of my Hip Op, it’s time for some basic tuition – though June, the sixth month, is still an inspiration, leading me to revisit, Lattice Labyrinths on the square lattice when both separation parameters are even – a situation I’ve been ignoring, as the Chinese and Serpentine families on the square lattice are completely soluble and the families of tessellations on the triangular lattice are so pretty.

When the separation parameters (a,b) are one even or zero, one odd, we get a Chinese Lattice Labyrinth such as (5,0) or (7,0). The area of the supertile a²+b² is always of the form 1+4n which corresponds to a tetragonally symmetrical supertile having a single central square and four identically-shaped arms. (A reminder: The separation parameters tell you how far across (a) and up (b) the lattice you have to count to come to a point identical in its environment to the one you started from; they tell you the scale on which the pattern repeats itself.)

The missing links graphs are simple to discover, the tessellation graphs use every lattice link not used in the missing links graphs (see my workbook).

If both a and b are even, a²+b² is of form 4+4n, so if a tessellation of tetragonally symmetrical supertiles is to exist, each must have at a block of at least four square tiles at the centre of each supertile. Finding such tessellations presents a big problem to me, as the missing-links graph algorithm as used above is of little help. Lower-order cases can be found by trial and error – below I illustrate case (6,4) – two different constructions.

I call the family of which (6,4) is an example the Windmill lattice Labyrinths, a name suggested by the shape of the lowest-order member of the family. I constructed the left-hand, yellowandgreentessellation by trial and error. Its tessellation graph is drawn in black. When one “reverse-engineers” this tessellation by constructing its missing-links graph , shown in red, one discovers that the result is the tessellation graph of another version of Windmill Lattice Labyrinth (6,4), shown on the right in yellow and blue.

This property, that a Windmill Labyrinths is “complementary” to another Windmill Labyrinth is shared by all members of the family. Some members are complementary to themselves, some are complementary to a mirror image of themselves, the rest to a differently-shaped tessellation, as in the case illustrated above.

Despite the above beautiful property, I find the Windmill family unsatisfactory because the lattice point at the centre of each supertile is unconnected by/to the tessellation graph. We CAN connect these points without upsetting the symmetry of the tessellation as a whole by constructing a cross at each supertile centre to divide each supertile into four identically-shaped arms. This does however mean that these tiles meet in edges of the tessellation graph rather than at vertex (lattice) points, corner to corner. This feels like a generalisation too far. However, this is not the end of the story for (even,even) separation parameters.

In the last few years a Japanese academic team (see reference at the end of the post) have been working their way through all the low-order polyominoes that tessellate, by an ingenious method quite different from mine and soon becoming impractical within the times of the order of the apparent age of the Universe for higher-order cases. Working through their results and correlating them with my lattice Labyrinth Families, I found one that didn’t fit! Here it is.

Well! Both separation parameters are even, the full symmetry of tetrad and dyad axes is there, the supertiles meet corner to corner AND all lattice points (corners of the squares of the square lattice) are connected in the tessellation graph. the above tessellation is indeed the lowest-order member of yet another infinite family, which I had to call the Japanese Lattice Labyrinths in honour of its source – and to balance nomenclaturely with the Chinese Lattice Labyrinth family.

(R, the repeat unit (or fundamental domain) is the number of squares in the pattern that are repeated to form the tessellation, S is the area of each supertile. In this case four supertiles in the four possible different orientations make up the repeat unit)

To return to June, month 6, here is another member of the Japanese family, (6,4).

Unlike the Windmill Labyrinths, the Japanese family can be constructed via a straightforward missing-links graph (shown here in red), though spotting general construction rules that cover all cases is not easy. However, my attempts to construct a Japanese Lattice Labyrinth for another “June” case, (6,2) met with persistent failure. It can’t be done! here is the best I can do:

I decided that Scarthin had to get an entry into my terminology; it’s where I live and work (and well worth a visit to the bookshop, Scarthin Books, the pub, The Boat Inn or just to relax on The Prom, overlooking the millpond, The Dam). The Scarthin Lattice Labyrinth family is undoubtedly also infinite in membership; its lowest-order “basic tessellation” is case (4,0), which you might like to try constructing. Alas, this family, like the Windmills has a block of four squares in each supertile with the central lattice point not connected to the tessellation graph.

We can characterise which (even,even) cases are drawable as Japanese Labyrinths and which as Scarthin labyrinths algebraically, but it’s perhaps easier to get the point from this little sample table, below.

At present I’ve forgotten my clear and simple explanation of why the (even,even) separation parameter cases fall into these two distinct families. Analogously to Arthur Sullivan’s musical “lost chord”; this is my “lost proof”. Seated one day at the laptop, I was fluent and free of fear…..

If the constructions above have left you baffled, befuddled and bewildered, then you could always purchase a copy of the explanatory workbook and be admitted to hours of recreational mathematics and artistic designing. The Lattice Labyrinths workbook is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

I am old enough to remember the celebration of the Coronation of Queen Elizabeth II, to whom we refer always as simply “The Queen”. I can even remember the date – 2nd. June 1953. In the village of Spratton, Northamptonshire, each of the pubs hosted a party. The Fir Tree Inn (long since closed) put on a barn dance, which sounded magic to me, but we, the village schoolmaster’s family, were too respectable for that; we went to the party in the village hall (alcohol-free? I wonder). I can just about remember watching the procession and service on a tiny black-and-white television screen. Afterwards we had the classic village hall tea – triangular sandwiches, probably including salmon and cucumber (Mmmm!), butterfly buns. iced cake and jelly or trifle. There was a children’s obstacle race, which included the terrifying task of blowing up a balloon until it burst. In 1977, for the Silver Jubilee of the Queen’s actual accession (not of the Coronation) I closed the young, but already full-time Scarthin Bookshop and joined in Cromford, Derbyshire’s big Street Party, the tables stretching the length of North Street. The Golden Jubilee in 2002 seemed to pass without a village celebration, or perhaps I was too busy to notice, but by 2012 I was heavily involved in the Celebrating Cromford organisation and helped put on a Diamond Jubilee Street Party on Scarthin Promenade, next to the Boat Inn. We mostly dressed up as queens or kings and there was music and an excellent children’s magician entertainer. The rain stopped and the sun came out just for that afternoon. Here’s a picture or two.

Anyway, now to the serious business of some Loyal Lattice Labyrinth Tessellations. The 21st. of April leads to the number pair (21,4), which specifies Trefoil Labyrinth (21,4). here it is.

I think this has turned out very appropriately for the Queen, it’s a self-disciplined tessellation, tightly controlled, resisting temptations to go out on limbs.

It’s also possible to dedicate a Lattice Labyrinth to all those reaching ninety years of age. 90 factorises to powers of primes: 2 x 5 x 3². We can immediately tell that 90 cannot be a Loeschian number of the form a² + ab + b² because such numbers and their prime power factors cannot be of the form 3n+2, which both 2 and 5 are. This rules out a tessellation on the triangular lattice with tile area 90. However, this number CAN be the sum of two squares because none of it’s prime power factors are of the form 4n+3 so it can correspond to a tessellation on the square lattice. We can quickly spot that 90 = 9²+ 3², yielding number pair (9,3) which as both are odd corresponds to Serpentine Lattice Labyrinth (9,3) and here it is, not in Union Jack colours this time. For once I’ve included some of the mechanics – the axes of symmetry and the missing-links graph employed in the construction.

Each supertile contains 45 squares, but they come in two sets, orientated at 90° to each other, so the repeat unit (or fundamental domain) is 90 as desired.

If the construction above has aroused your curiosity, then you could always purchase a copy of the explanatory workbook and be admitted to hours of recreational mathematics and artistic designing. The Lattice Labyrinths workbook is available from the publisher or you-know-who , or from a good independent bookshop or via Google.

The question of whether or not Britain should be a member of the EU is too complicated and subtle even for ME, let alone the average British elector to decide upon. It is JUST the sort of question we elect a parliament of professional legislators, able to commission a wide range of expert advice, to determine. But, ironically, these elected savants have determined to leave it up to us, the ill-informed electorate, to decide, subject as we are to the winds of contingent events and the plausible posturing of high-vis rogues on both sides. I and many of my acquaintances despair of the wisdom, even of the goodwill, of our government. Actually, this is my permanent state of mind.

Well, if we must have this stupid referendum, at least we should follow the usual constitution of societies and public bodies, for instance that internationally insignificant Derbyshire village body, the Cromford Community Association and require a two-thirds to one-third majority of members (and a simple majority of the Council ) before taking the momentous step of changing our long-established Constitution. What is the UK to do if (as is quite likely) the vote is 51% to 49% in either direction. Should such a divided nation LEAVE despite a tiny majority in favour of STAYING? Should a two-thirds/one third majority be required for us to stay IN, taking OUT to be the historical default situation?

Afterword; as you may know, the vote was 52% to 48% – little different to my “forecast” and so close I’m not bothering to confirm which way it went, though a statistician would confirm that such a split of some 34 million votes means there is a “highly significant” deviation from a 50:50 coin-toss hypothesis. One reaction to this result can be enjoyed on my twitter page: https://twitter.com/davescarthin/status/747816380930596864

Anyway, in an attempt to hear some substantial arguments, rather than the national media’s head-count of moguls, Scarthin Books of Cromford are sponsoring an IN/OUT debate on Saturday 9th. Aprilat the aforementioned Cromford Community Centre. The contestants are Edward Spalton and Brian Mackenzie, both experienced in international enterprise and trade,both committed to their views beyond the economic arguments, both worthy of respect. Here is the poster, and also a parking/access plan as our industrious village which, with its shops, hotels, stone quarries, engineering and service businesses,historic mills and housing, restaurants and pubs, is a HUB and often congested.

Of course, Britain will probably be OK either OUT or IN the EU. We’ll all try to make the best of whatever situation we find ourselves in. The result of the referendum will be subject to huge random (i.e. inexplicable and unpredictable) factors; we might just as well have tossed a coin – HEADS WE’RE IN, TAILS WE’RE OUT – you could say that tossing a coin is exactly what we ARE doing.

One afternoon in the the Chemistry Laboratory at Northampton Grammar School, a teacher stuck a pin into a list of Cambridge Colleges and thus randomly allocated a College to each of us five Science Boys. The 100% success record of Alan Bennett’s History Boys was not achieved; the two most promising did not get in (they shone at Imperial College, London, instead), three of us did – to Pembroke, St Catherine’s and St. John’s. Had I not found myself at John’s, I would not, on my way to Cavendish Laboratory lectures, have passed and become addicted to David’s bookstall – and Scarthin Books, my children and this blog would all vanish “in a puff of smoke”. But surely another life, not empty of colour and creation, would have come about in place of the life I’ve actually been living. So will it be with Britain, in or out of the EU.

WHAT ABOUT A TESSELLATION ?! The EU referendum takes place on June 23rd., so here is Trefoil Lattice Labyrinth (23,6), first of all the usual pretty pattern of just six supertiles, each comprised of 23² + 23×6 + 6² = 703 equilateral triangles. I’ve produced it in the EU flag colours. You could call it the Brussels Labyrinth, Brussel Labyrint, Le Labyrinthe de Bruxelles.

BUT, there are 28 nations in the EU. Here they are in all their waning and waxing variety fully interlocked and symmetrically disposed, three nines, about one central supertile (I wonder who that can be?). If the UK leaves, the symmetry will certainly be destroyed